Multi-period portfolio optimization in a defined contribution pension plan during the decumulation phase

Chuangwei Lin; Li Zeng; Huiling Wu

doi:10.3934/jimo.2018059

Article Contents

2019, Volume 15, Issue 1: 401-427. Doi: 10.3934/jimo.2018059

This issue Previous Article A class of two-stage distributionally robust games Next Article

Multi-period portfolio optimization in a defined contribution pension plan during the decumulation phase

1.
Research Center for International Trade and Economic, Guangdong University of Foreign Studies, Guangzhou 510006, China
2.
School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China
3.
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

* Corresponding author: zsuzengli@126.com
* Corresponding author: zsuzengli@126.com

Received: September 1, 2017

Revised: December 1, 2017

Early access: April 21, 2018

Published online: January 1, 2019

The first author is supported by National Natural Science Foundation of China (No. 11671411) and Innovative School Project in Higher Education of Guangdong, China (No. GWTP-SY-2014-02).

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Abstract

This paper studies a multi-period portfolio selection problem for retirees during the decumulation phase. We set a series of investment targets over time and aim to minimize the expected losses from the time of retirement to the time of compulsory annuitization by using a quadratic loss function. A target greater than the expected wealth is given and the corresponding explicit expressions for the optimal investment strategy are obtained. In addition, the withdrawal amount for daily life is assumed to be a linear function of the wealth level. Then according to the parameter value settings in the linear function, the withdrawal mechanism is classified as deterministic withdrawal, proportional withdrawal or combined withdrawal. The properties of the investment strategies, targets, bankruptcy probabilities and accumulated withdrawal amounts are compared under the three withdrawal mechanisms. Finally, numerical illustrations are presented to analyze the effects of the final target and the interest rate on some obtained results.

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Mathematics Subject Classification: Primary: 91G10, 91G80; Secondary: 90C90, 91B02.

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Figure 1. F_n over time

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Figure 2. The targets and the expected wealths over time

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Figure 3. ${\rm{E}}_{0, x_0}\left((1-\rho_n)X_n-c_n\right)$ and ${\rm{E}}_{0, x_0}(\pi_n^*)$ over time

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Figure 4. Accumulated withdrawal amounts over time

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Figure 5. Expected withdrawal amounts at each time

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Figure 6. $F_n-{\rm{E}}_{0, x_0}(X_n), ~n = 0, 1, \ldots, T$

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Figure 7. Expectation of the accumulated withdrawal amount over time

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Figure 8. Growth ranges of the accumulated withdrawal amounts

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Table 1. Frequencies of some events that $X_T$ is in the neighborhood of $F_T$

$X_T\in[a, b)$	Deter. withdrawal Frequencies	Comb. withdrawal Frequencies	Prop. withdrawal Frequencies
$(-\infty, F_T-6000)$	0.3024	0.2472	0.1985
$[F_T-6000, F_T-4000)$	0.0997	0.0885	0.0791
$[F_T-4000, F_T-2000)$	0.1811	0.1801	0.1724
$[F_T-2000, F_T)$	0.4168	0.4842	0.5500
$[F_T, +\infty)$	0	0	0

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Table 2. The total number of bankruptcies-10000 simulations

Events	Deter. withdrawal Number of simulations	Comb. withdrawal Number of simulations	Prop. withdrawal Number of simulations
$S=0$	9692	9726	9729
$S\in(0, 50]$	265	247	246
$S\in(50,100]$	39	23	22
$S\in(100,150]$	4	4	3
$S\in(150, +\infty)$	0	0	0
	Deter. withdrawal	Comb. withdrawal	Prop. withdrawal
Mean of $S$	0.7341	0.5870	0.5060

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Table 3. The first occurrence of bankruptcy

Events	Deter. withdrawal Number of simulations	Comb. withdrawal Number of simulations	Prop. withdrawal Number of simulations
$\tau\in[0, 60)$	100	117	153
$\tau\in[60,120)$	142	118	92
$\tau\in[120 180]$	66	39	26
	Deter. withdrawal	Comb. withdrawal	Prop. withdrawal
Mean of $\tau$	86	76	65

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Table 4. The mean of $\tilde \tau$ and $\tilde S$-10000 simulations

	Deter. withdrawal	Comb. withdrawal	Prop. withdrawal
Mean of $\tilde \tau$	64.4634	51.1655	41.9057
Mean of $\tilde S$	24.1194	21.6262	18.9368

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Table 5. Accumulated withdrawal amounts and their relative changes

Time	Deter. withdrawal	Comb. withdrawal	Prop. withdrawal
$n=36$	027,922	028,338	029,112
$n=72$	055,088 (27166)	056,223 (27885)	058,294 (29182)
$n=108$	082,255 (27167)	083,865 (27642)	086,875 (28581)
$n=144$	109,422 (27167)	110,911 (27046)	114,109 (27234)
$n=180$	136,589 (27167)	137,134 (26223)	139,640 (25531)

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Table 6. The total number of bankruptcy-10000 simulations

	Events	Deter. withdrawal	Comb. withdrawal	Prop. withdrawal
$r^f=1.0020$	$S=0$	9672	9697	9701
$r^f=1.0020$	Mean of $S$	0.8268	0.6933	0.6123
$r^f=1.0025$	$S=0$	9692	9726	9729
$r^f=1.0025$	Mean of $S$	0.7341	0.5870	0.5060
$r^f=1.0030$	$S=0$	9690	9724	9718
$r^f=1.0030$	Mean of $S$	0.6565	0.5304	0.5001

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Table 7. The total number of bankruptcies-10000 simulations

	Events	Deter. withdrawal	Comb. withdrawal	Prop. withdrawal
$F_T=1.4 (x_0/ä_{60})_{75}$	$S=0$	9771 simulations	9820 simulations	9850 simulations
$F_T=1.4 (x_0/ä_{60})_{75}$	Mean of $S$	0.4555	0.3011	0.2060
$F_T=1.5 (x_0/ ä_{60})ä_{75}$	$S=0$	9692 simulations	9726 simulations	9729 simulations
$F_T=1.5 (x_0/ ä_{60})ä_{75}$	Mean of $S$	0.7341	0.5870	0.5060
$F_T=1.6$$(x_0/ä_{60})ä_{75}$	$S=0$	9608 simulations	9611 simulations	9540 simulations
$F_T=1.6$$(x_0/ä_{60})ä_{75}$	Mean of $S$	0.9601	0.8639	0.8636

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